AE doesn't separate audio. You'll have to do it elsewhere, possibly in Premiere Pro or tools like ffmpeg/MPEGStreamclip. It's odd, though. since these components are the same in all the suite apps, a file that works in AE should just the same open in AU (without showing teh video in teh worst case). .mts is just MPEG2 or AVCHD in disguise, anyway, which the apps should open without a hitch - if it's not out of spec. I'd look into your camera settings, though. maybe you recorded to the wrong channels or an odd audio sampel rate and it's messing things up.

Did Adobe take functionality away from us with the Audition release? I used to be able to open up all of my video files in Soundbooth, clean up the audio and then re-save them before pushing them into P-Pro, After Effects, or even Captivate.

You will have to ask that on the AU forum. While it has inherited SB's processing routines, indeed some things will work considerably differently. I share your sentiment, SB was much more straightforward for us not-so-audiophiles. A pity it got killed of in favor of a more convoluted app....